The use of heat exchangers for cooling hot liquid from a pump chamber which flows along the pump shaft to a mechanical seal assembly associated with the pump is well known in the art. One such arrangement is illustrated and described in my prior U.S. Pat. No. 3,459,430, granted July 6, 1967. The heat exchanger described in this patent includes a pair of concentric coils one for liquid flowing to the stuffing box and the other for liquid flowing through a pressure-breakdown means associated with the mechanical seals. The coils are contained within in concentric jackets which communicate with one another at the lower ends. A coolant, such as water or other fluid, is introduced into the top of the outermost jacket, so as to flow downwardly and over the coils in that jacket, through the lower end of the innermost jacket and then out the top of the innermost jacket. Cooling of the liquid in the coils is accomplished by mass flow of coolant into and out of the jackets; cooling may be non-uniform because of the irregular paths of flow available to the coolant; there may be hot spots which receive little or no cooling. Further, the heat exchanger of the prior art patent will provide cooling for all mechanical seals only in running condition; when the pump assembly is in hot, stand-by condition, the bottom seals are not cooled while the top seals are cooled, because of the requirements for a recirculating impeller for the heat exchanger.